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Definition of Differentiation

Sarah Miller

Sarah Miller

5 min read

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Study Guide Overview

This study guide covers instantaneous rate of change, focusing on its definition as the slope of a graph at a specific point. It explains how to calculate it using limits and its relationship to the derivative. Examples, practice questions, and a glossary of key terms (including average rate of change) are provided.

Instantaneous Rate of Change

Table of Contents

  1. What is the Instantaneous Rate of Change?
  2. Finding the Instantaneous Rate of Change
  3. Worked Example
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

What is the Instantaneous Rate of Change?

The instantaneous rate of change is the slope of a graph at a specific point, rather than between two points.

Key Concept

The instantaneous rate of change at a point on a function is essentially the derivative at that point. It measures how the function value changes as the input changes infinitesimally.

Finding the Instantaneous Rate of Change

Consider the graph of a function ff with two points on the graph, AA and BB:

Average Rate of Change

Using the labeling on the left:

  • The average rate of change from AA to BB can be written as: f(x)f(a)xa\frac{f(x) - f(a)}{x - a}

Using the labeling on the right:

  • T...